Optimal control for a fractional tuberculosis infection model including the impact of diabetes and resistant strains

Journal of Advanced Research 17 (2019) 125–137

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Journal of Advanced Research
journal homepage: www.elsevier.com/locate/jare

Original article

Optimal control for a fractional tuberculosis infection model including
the impact of diabetes and resistant strains
N.H. Sweilam a,⇑, S.M. AL-Mekhlafi b, D. Baleanu c,d
a

Cairo University, Faculty of Science, Mathematics Department, 12613 Giza, Egypt
Sana’a University, Faculty of Education, Mathematics Department, Sana’a, Yemen
c
Cankaya University, Department of Mathematics, 06530, Ankara, Turkey
d
Institute of Space Sciences, P.O. Box MG 23, Magurele, 077125 Bucharest, Romania
b

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

 Optimal control problem for the

fractional TB infection model is
presented.
 The nonstandard two-step Lagrange

interpolation method is presented for
numerically solving the optimality
system.
 Necessary and sufficient conditions
that guarantee the existence and the
uniqueness of the solution of the
control problem are given.
 Four controls variables are proposed
to minimize the cost of interventions.
 New numerical schemes for
simulating fractional order optimality
system with Mittag-Leffler kernel are
given.

a r t i c l e

i n f o

Article history:
Received 3 November 2018
Revised 22 December 2018
Accepted 13 January 2019
Available online 19 January 2019
Keywords:
Tuberculosis model
Diabetes and resistant strains
Atangana-Baleanu fractional derivative
Lagrange polynomial interpolation
Nonstandard two-step Lagrange
interpolation method

a b s t r a c t
The objective of this paper is to study the optimal control problem for the fractional tuberculosis (TB)
infection model including the impact of diabetes and resistant strains. The governed model consists of
14 fractional-order (FO) equations. Four control variables are presented to minimize the cost of interventions. The fractional derivative is defined in the Atangana-Baleanu-Caputo (ABC) sense. New numerical
schemes for simulating a FO optimal system with Mittag-Leffler kernels are presented. These schemes
are based on the fundamental theorem of fractional calculus and Lagrange polynomial interpolation.
We introduce a simple modification of the step size in the two-step Lagrange polynomial interpolation
to obtain stability in a larger region. Moreover, necessary and sufficient conditions for the control
problem are considered. Some numerical simulations are given to validate the theoretical results.
Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Cairo University. This is an open access article
under the CC BY-NC-ND license ( />
Peer review under responsibility of Cairo University.
⇑ Corresponding author.
E-mail address: (N.H. Sweilam).
/>2090-1232/Ó 2019 The Authors. Published by Elsevier B.V. on behalf of Cairo University.
This is an open access article under the CC BY-NC-ND license ( />

126

N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

Introduction
A new study suggests that millions of people with high blood
sugar may be more likely to develop tuberculosis (TB) than previously expected. TB is a severe infection that is caused by bacteria in
the lungs and kills many people each year, in addition to HIV/AIDS
and malaria, according to the Daily Mail website [1]. In 2017,
according to the World Health Organization nearly 10 million people were infected with TB [2]. Experts are concerned that a global
explosion in the number of diabetes cases will put millions of people at risk [3].
Many mathematical models have been proposed to elucidate

the patterns of TB [4–7], Recently, Khan et al., [8], presented a
new fractional model for tuberculosis. In addition, several papers
considered modeling TB with diabetes; see, for example, [9–12].
Recently, Carvalho and Pinto presented non-integer-order analysis
of the impact of diabetes and resistant strains in a model of TB
infection [13]. Fractional-order (FO) models provide more accurate
and deeper information about the complex behaviors of various
diseases than can classical integer-order models. FO systems are
superior to integer-order systems due to their hereditary properties and description of memory [14–28]. Fractional optimal control
problems (FOCPs) are optimal control problems associated with
fractional dynamic systems. Fractional optimal control theory is a
very new topic in mathematics. FOCPs may be defined in terms
of different types of fractional derivatives. However, the most
important types of fractional derivatives are the RiemannLiouville and Caputo fractional derivatives [29–40]. In addition,
the theory of FOCPs has been under development. Recently, some
interesting real-life models of optimal control problems (OCPs)
were presented elsewhere [41–52].
A new concept of differentiation was introduced in the literature whereby the kernel was converted from non-local singular
to non-local and non-singular. One of the great advantages of this
new kernel is its ability to portray fading memory as well as the
well-defined memory of the system under investigation. A new
FO derivative, based on the generalized Mittag-Leffler function as
a non-local and non-singular kernel, was presented by Atangana
and Baleanu [14] in 2016. The newly introduced AtanganaBeleanu derivative has been applied in the modeling of various
real-world problems in different fields, as previously discussed
[15–22]. This derivative, based on the Mittag-Leffler function, is
more suitable for describing real-world complex problems.
Numerical and analytical methods are very useful because they
can play very necessary roles in characterizing the behavior of
the solution of the fractional differential equations, as shown in

[15–27].
To the best of our knowledge, the optimal control for a FO
tuberculosis infection model that includes diabetes and resistant
strains has never been explored. The main contribution of this
work is to propose a class of FOCPs and develop a numerical
scheme to provide an approximate solution for those FOCPs. We
consider the mathematical model in Khan et al. [8], and the fractional derivative is defined here in the Atangana-Baleanu-Caputo
(ABC) sense. A new generalized numerical scheme for simulating
a FO optimal system with Mittag-Leffler kernels is established.
These schemes are based on the fundamental theorem of fractional
calculus and Lagrange polynomial interpolation. This paper was
organized as follows. Fundamental relations are given in ‘‘Fundamental Relations”. In ‘‘Fractional Model for TB Infection Including
the Impact of Diabetes and Resistant Strains”, the fractionalorder model with four control variables is introduced. The proposed control problem with the optimality conditions is given in
‘‘Formulation of the Fractional Optimal Control Problem”. In
‘‘Numerical Techniques for the Fractional Optimal Control Model”,

numerical schemes with exponential and Mittag-Leffler laws are
presented. Numerical experiments are given in ‘‘Numerical Simulations”. In ‘‘Conclusions”, the conclusions are presented.
Fundamental relations
In the following, the basic fractional-order derivative definitions
used in this paper are given.
Definition 1. The Liouville-Caputo FO derivative is defined as in
[53]:
C a
a Dt g ðt Þ

¼

1
Cð1 À aÞ

Z

t

ðt À qÞÀa g_ ðqÞdq; 0 < a

ð1Þ

1:

0

Definition 2. The Atangana-Baleanu fractional derivative in the
Liouville-Caputo sense is defined as in [14]:
ABC a
a Dt gðtÞ

¼

BðaÞ
ð1 À aÞ

Z

t

ðEa ðÀa

ðt À qÞa

_
ÞgðqÞdq
;
ð1 À aÞ

ð2Þ

0

where BðaÞ ¼ 1 À a þ CðaaÞ is the normalization function.
Definition 3. The corresponding fractional integral concerning the
Atangana–Baleanu-Caputo derivative is defined as [14]
ABC a
a It gðtÞ

¼

ð1 À aÞ
a
gðtÞ þ
BðaÞ
BðaÞCðaÞ

Z

t

_
ðt À qÞaÀ1 gðqÞdq;

0

They found that when a is zero, they recovered the initial function, and if a is 1, they obtained the ordinary integral. In addition,
they computed the Laplace transform of both derivatives and
obtained the following:
a
fABC
0 Dt g ðt Þg ¼

BðaÞGðpÞpa À paÀ1 gð0Þ
ð1 À aÞðpa þ ð1Àa aÞÞ

Theorem 1. For a function g 2 C [a, b], the following result holds [9]:
a
jjABC
a Dt g ðt Þjj <

BðaÞ
jjg ðt Þjj; where jjg ðt Þjj ¼ maxa
ð1 À aÞ

t b jg ðt Þj;

Further, the Atangana–Baleanu-Caputo derivatives fulfill the
Lipschitz condition [9]:
a
ABC a
jjABC
a Dt g 1 ðt Þ À a Dt g 2 ðt Þjj <

-jjg 1 ðtÞ À g 2 ðtÞjj

Fractional model for TB infection including the impact of
diabetes and resistant strains
In this section, we study fractional optimal control for TB infection including the impact of diabetes and resistant strains, as given
in Carvalho and Pinto [13]. So that the reader can make sense of the
model, Fig. 1 shows the flowchart of the model as given in Carvalho
and Pinto [13]. The fractional derivative here is defined in the ABC
sense. We add four control functions, u1 , u2 , u3 and u4 ; and four real
positive model constants, xi ; i ¼ 1; 2; 3; 4 and xi 2 ð0; 1Þ. These
controls are given to prevent the failure of treatment in I1s , I1R , I2s
and I2R , e.g., patients’ health care providers encourage them to complete the treatments by taking TB and diabetes medications regularly. This model consists of fourteen classes. Let us consider the
population to be divided into diabetic (index 1) and non-diabetic


127

N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

Fig. 1. Flowchart of the model [13].

(index 2). Then, we have susceptible individuals (S2 and S1 ), individuals exposed and sensitive to TB (E2s and E1s ), individuals exposed
and resistant to TB (E2R and E1R ), individuals infected with and sensitive to TB (I2s and I1s ), individuals infected with and resistant to TB
(I2R and I1R ), individuals recovering from and sensitivite to TB (R2s
and R1s ), and individuals recovering from and resistant to TB (R2R
and R1R ). All the parameters for the modified model in Table 1,
depend on the FO because the use of the constant parameter a
instead of an integer parameter can lead to better results, as one
has an extra degree of freedom [40]. The main assumption of this
model is that the total population N is a constant in time, i.e., the

a
a
birth and death rates are equal and d1 ¼ d2 ¼ 0. The resulting model
with four controls is given as follows:
ABC a
a Dt S1

¼ Aa À ðla þ aaD þ kT ÞS1 ;

ð3Þ

ABC a
a Dt S2

¼ aaD S1 À ðla þ hkT ÞS2 ;

ð4Þ

ABC a
a Dt E1s

¼ nð1 À P1 ÞkT S1 þ nE1s þ r32 ð1 À da1 kT R1R Þ À ð1 À r a1 Þ
ð6Þ

À
Á
¼ ð1 À nÞð1 À P2 ÞhkT S2 þ r41 1 À da2 hkT R2s þ aaD E1s
a

À ð1 À ra2 Þðk2 þ r2 hkT ÞE2s À ðn þ la ÞE2s ;

ð8Þ

a

a

À ðs1 aaD þ g1 n þ ca11 þ la þ d1 þ x1 u1 ÞI1s ;
a

À ðda11 þ n þ aaD þ la ÞR1s
ABC a
a Dt R1R

ð13Þ

¼ ca21 I1R þ x2 u2 I1R þ nR1s À r32 ð1 À da1 kT R1R Þ
À ðda12 þ aaD þ la ÞR1R ;

ABC a
a Dt R2s

ABC a
a Dt R2R

ð12Þ

ð14Þ

¼ ca21 I2s þ x3 u3 I2s þ aaD R1s À r41 hð1 À da2 ÞkT R2s

ð15Þ

¼ ca22 I2R þ x4 u4 I2R þ nR2s þ aD R1R
À
Á
À r42 h 1 À da2 kT R2R À ðda22 þ la ÞR2R

ð16Þ

where

I1s þ eI1R þ e1 I2s þ e2 I2R
N

Let us consider the state system presented in Eqs. (3)–(16), in
R14 ; with the set of admissible control functions

X ¼ fðu1 ð:Þ; u2 ð:Þ; u3 ð:Þ; u4 ð:ÞÞjui is Lebsegue measurable on ½0; 1Š;
0

¼ nP1 kT S1 þ ð1 À r1 Þðk1 þ r1 kT ÞE1R þ g1 nI1s þ d12 R1R
a

a

¼ ca11 I1s þ x1 u1 I1s À r31 ð1 À da1 ÞkT R1s

ð9Þ
a

À ðs1 aaD þ ca12 þ la þ d1 þ x2 u2 ÞI1R

a

Control problem formulation

¼ ð1 À nÞP1 kT S1 þ ð1 À r a1 Þðk1 þ r1 kT ÞE1s þ da11 R1s

a

a

þ d22 R2R À ðc22 þ l þ d2 þ x4 u4 ÞI2R ;

kT ¼ b

À
Á
¼ nð1 À P2 ÞhkT S2 þ nE2s þ r42 1 À da2 hkT R2R þ aaD E1R
a

ABC a
a Dt I1R

a

¼ nP 2 hkT S2 þ ð1 À r a2 Þðk2 þ r2 hkT ÞE2R þ g2 nI2s þ s1 aaD I1R
a

ABC a

a Dt R1s

ð11Þ

ð7Þ

À ð1 À ra2 Þðk2 þ r2 hkT ÞE2R À la E2R ;
ABC a
a Dt I1s

ABC a
a Dt I 2R

À ðd21 þ n þ la ÞR2s ;
ð5Þ

a

ABC a
a Dt E2R

a

þ da21 R2s À ðg2 n þ ca21 þ la þ d2 þ x3 u3 ÞI2s ;

a

 ðk1 þ r1 kT ÞE1R À ðaaD þ la ÞE1R ;
ABC a
a Dt E2s

À
ÁÀ a
Á
¼ ð1 À nÞP2 hkT S2 þ 1 À r a2 k21 þ r2 hkT E2s þ s1 aaD I1s

¼ ð1 À nÞð1 À P1 ÞkT S1 þ r31 ð1 À da1 R1s Þ À ð1 À r a1 Þðk1
þ r1 kT ÞE1s À ðn þ aaD þ la ÞE1s ;

ABC a
a Dt E1R

ABC a
a Dt I 2s

ð10Þ

u1 ð:Þ; u2 ð:Þ; u3 ð:Þ; u4 ð:Þ

where T f is the final
are controls functions:

Â
Ã
1; 8t 2 0; T f ; i ¼ 1; 2; 3; 4g;
time

and

u1 ð:Þ; u2 ð:Þ; u3 ð:Þ and u4 ð:Þ

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N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

subject to the constraint

Table 1
The parameters of systems (3)–(16) and their descriptions [13].
Parameter

Descriptions

Values

Aa

Recruitment rate
Diabetes acquisition rate

667685.

aaD
ba

ABC a
a Dt S1

ABC a

a Dt E1R

9
Àa
1000 yr

Effective contact rate for TB infection
Modification parameter
Modification parameter
Modification parameter
Modification parameter
Rate of natural death

f5; 8; 9g
1:1
1:1
1:1
2
0:04
0:03

a

d1

Rate of TB infection among diabetic individuals
Rate of TB infection among non-diabetic
individuals
Rate of TB infection among diabetic individuals
Non-diabetic individuals’ chemoprophylaxis rate

Diabetic individuals’ chemoprophylaxis rate.
Non-diabetic individuals’ degree of immunity
Diabetic individuals’ degree of immunity
Non-diabetic individuals’ rate of endogenous
reactivation
Diabetic individuals’ rate of endogenous
reactivation
Non-diabetic individuals’ sensitive TB infection
recovery rate
Non-diabetic individuals’ resistant TB infection
recovery rate
Diabetic individuals’ sensitive TB infection
recovery rate
Diabetic individuals’ resistant TB infection
recovery rate
Rate of death due to TB

d2

a

Rate of death due to TB and diabetes

0yrÀa

s1
g1
g2

Modification parameter

Modification parameter
Modification parameter
Non-diabetic individuals of partial immunity
Non-diabetic individuals’ partial immunity for
sensitive recovered
Non-diabetic individuals’ partial immunity after
resistant recovery
Diabetic individuals’ of partial immunity
Sensitive recovered diabetic individuals’ partial
immunity
Resistant recovered diabetic individuals’ partial
immunity
Sensitive recovered non-diabetic individuals’
degree of immunity
Resistant recovered non-diabetic individuals’
degree of immunity
Sensitive recovered diabetic individuals’ degree of
immunity
Recovered diabetic individuals’ degree of
immunity

1:01
1:01
1:01
0:0986yr Àa
0:0986yr Àa

ea
ea1
ea2

ha

la
n
P1
P2
r a1
r a2

r1
r2
a

k1

a

k2

ca11
ca12
ca21
ca22

da1
da11
da12
da2
da21

ca22

ra31
ra32
ra41
ra42

1
Àa
53:5 yr

Z

Tf

ðI1s þ I1R þ I2s þ I2R þ

B3 2
B4
u ðtÞ þ u24 ðtÞÞdt;
2 3
2

Z

Tf

¼ n7 ;

ABC a
a Dt I 2R

¼ n10 ;

ABC a
a Dt R1s

ABC a
a Dt R2s

¼ n13 ;

ABC a
a Dt R2R

ABC a
a Dt I 2s

¼ n8 ;

¼ n11 ;

¼ n6 ;

¼ n9 ;

ABC a
a Dt R1R

¼ n12 ;

¼ n14 ;

2K 1 yr Àa

R1s ð0Þ ¼ R1s0 ; R1R ð0Þ ¼ R1R0 ; R2s ð0Þ ¼ R2s0 ; R2R ð0Þ ¼ R2R0 :

0:7372yrÀa

To define the FOCP, consider the following modified cost function [31]:
Z Tf
$
J¼
½Ha ðS1 ;S2 ;E1s ;E1R ;E2s ; E2R ;I1s ;I1R ;I2s ; I2R ;R1s ;R1R ;R2s ; R2R ;uj ;tÞ

0:7372yrÀa
0:7372yrÀa

i ¼ 1; :::; 14; and the following initial conditions are satisfied:

S1 ð0Þ ¼ S01 ; S2 ð0Þ ¼ S02 ; E1s ð0Þ ¼ E1s0 ; E1R ð0Þ ¼ E1R0 ; E2s ð0Þ ¼ E2s0 ;
E2R ð0Þ ¼ E2R0 ; I1s ð0Þ ¼ I1s0 ; I1R ð0Þ ¼ I1R0 ; I2s ð0Þ ¼ I2s0 ; I2R ð0Þ ¼ I2R0 ;

0

0:7372yrÀa
0yr

À

Àa

14
X
ðki ni ðS1 ;S2 ; E1s ;E1R ;E2s ;E2R ; I1s ;I1R ;I2s ; I2R ;R1s ; R1R ;R2s ; R2R ;uj ; tފdt;
i¼1

0:0986yr Àa
0:1yrÀa
0:1yrÀa

ð19Þ

where j ¼ 1; 2; 3; 4; and i ¼ 1; :::; 14.
The Hamiltonian is given as follows:

À
Á
Ha S1 ; S2 ; E1s ; E1R ;E2s ; E2R ;I1s ;I1R ; I2s ; I2R ;R1s ; R1R ;R2s ; R2R ;uj ; ki ;t
À
Á
¼ g S1 ; S2 ; E1s ;E1R ; E2s ; E2R ;I1s ; I1R ; I2s ;I2R ;R1s ; R1R ;R2s ; R2R ;uj ; t
þ

14
X

ki ni ðS1 ;S2 ;E1s ; E1R ;E2s ; E2R ; I1s ;I1R ;I2s ; I2R ; R1s ;R1R ; R2s ; R2R ;uj ;tÞ;

i¼1

ð20Þ

0:1yrÀa

where, j ¼ 1; 2; 3; 4; and i ¼ 1; :::; 14.
From Eqs. (19) and (20), the necessary and sufficient conditions
for the FOCP [34–37] are as follows:

0:73P 1
0:73P 1
0:71P 2
0:71P 2

B1 2
B2
u ðtÞ þ u22 ðtÞ
2 1
2
ð17Þ

gðS1 ; S2 ; E1s ; E1R ; E2s ; E2R ; I1s ; I1R ; I2s ; I2R ; R1s ; R1R ;

ABC a
Dtf k1
t

¼

@Ha
;
@S1

ABC a
Dtf k2
t

¼

@Ha
;
@S2

ABC a
Dtf k3
t

¼

@Ha
;
@E1s

ABC a
Dtf k4
t

¼

@Ha
;
@E1R

ABC a
Dtf k5
t

¼

@Ha
;
@E2s

ABC a
Dtf k6
t

¼

@Ha
;
@E2R

ABC a
Dtf k7
t

¼

@Ha
;
@I1s

ABC a
Dtf k8
t

¼

@Ha
;
@I1R

ABC a
Dtf k9
t

¼

@Ha
;
@I2s

ABC a
Dtf k10
t

ð18Þ

¼

¼

@Ha
;
@R1s

ABC a
Dtf k12
t

¼

@Ha
;
@R1R

ABC a
Dtf k13
t

¼

@Ha
;
@R2s

ABC a
Dtf k14

t

¼

@Ha
;
@R2R

0¼

@H
;
@uk

ð21Þ

@Ha
;
@I2R

ABC a
Dtf k11
t

0

R2s ; R2R ; u1; u2; u3; u4; tÞdt;

ABC a
a Dt I 1R

ABC a
a Dt E2R

¼ n5 ;

ABC a
a Dt I 1s

¼ n3 ;

where

where B1, B2, B3, and B4 are the measure of the relative cost of the
interventions associated with the controls u1, u2, u3, and u4.
Then, we find the optimal controls u1 ; u2 ; u3 and u4 that minimize the cost function

Jðu1 ; u2 ; u3 ; u4 Þ ¼

ABC a
a Dt E2s

¼ n4 ;

ABC a
a Dt E1s

¼ n2 ;

ni ¼ nðS1 ; S2 ; E1s ; E1R ; E2s ; E2R ; I1s ; I1R ;I2s ; I2R ; R1s ; R1R ; R2s ; R2R ; u1 ; u2 ; u3 ;u4 ; tÞ;

0

þ

ABC a
a D t S2

0:06
0yrÀa
0yrÀa
0:75P 1
0:7P 2
0:00013yr Àa

The objective function is defined as follows:

Jðu1 ; u2 ; u3 ; u4 Þ ¼

¼ n1 ;

ð22Þ


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N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137
ABC a
0 Dt S1

@Ha
;
@k1

¼

ABC a
0 D t S2

¼

@Ha
;
@k2

Àð1 À r a1 Þr1

b à Ã
b
b
E k þ ð1 À nÞð1 À P2 Þh SÃ2 kÃ5 þ r41 ð1 À da2 Þ RÃ2R kÃ5
N 1R 4
N
N

ABC a
0 Dt E1s

¼

@Ha
;
@k3

ABC a
0 Dt E1R

¼

@Ha
;
@k4

Àð1 À r a2 Þr2 hkT E2s kÃ5 þ nð1 À P2 Þh

ABC a
0 Dt E2s

¼

@Ha
;
@k5

ABC a
0 Dt E2R

¼

@Ha

;
@k6

Àð1 À r a2 Þr2 hkT E2R kÃ6 þ ð1 À nÞP1

¼

@Ha
;
@k7

ABC a
0 Dt I1s

@Ha
ABC a
;
0 Dt I2s ¼
@k9

¼

ABC a
0 Dt I 2R

@Ha
¼
;
@k10

@Ha
ABC a
;
0 Dt R1s ¼
@k11

ABC a
0 Dt R1R

@Ha
;
@k13

ABC a
0 Dt R2R

ABC a
0 Dt R2s

¼

@Ha
;
@k8

ABC a
0 Dt I 1R

a

Â

þð1 À r a2 Þr2 hE2R

@Ha
;
@k14
ð23Þ

are the Lagrange multipliers. Eqs. (21) and (22) describe the necessary conditions in terms of a Hamiltonian for the optimal control
problem defined above. We arrive at the following theorem:

þr42 ð1 À da2 Þ
ABC a Ã
Dtf k8
t

Theorem 2. Let SÃ1 , SÃ2 , EÃ1R , EÃ1s , EÃ2R , EÃ2s , IÃ1R , IÃ1s ; IÃ2s , IÃ2R ; RÃ1R ; RÃ1s ; RÃ2R ;
RÃ2s be the solutions of the state system and uÃi , i ¼ 1; Á Á Á ; 4 be the given
optimal controls. Then, there exists co-state variables kÃj ; j ¼ 1; Á Á Á ; 14
satisfying the following:

(i) Co-state equations:

¼ ðÀðla þaD þ kT ÞkÃ1 þ aaD k2 þ ðð1 À nÞð1 À P1 ÞkT ÞkÃ3
þ ðnð1 À P1 ÞkT ÞkÃ4 þ ðð1 À nÞP 1 kT ÞkÃ7 þ ðnP1 kT ÞkÃ8 Þ;
a

¼ ðÀðl þ

hkT ÞkÃ2

þ ðð1 À nÞð1 À

a

a

ð26Þ

a

¼ ðÀkÃ4 ð1 À r a1 Þðk1 þ r1 kT Þ À ðaaD þ la Þ þ kÃ6 aaD
a

þ kÃ8 ð1 À r a1 Þðk1 þ r1 kT ÞÞ;
ABC a Ã
Dtf k5
t

ð25Þ

¼ ðÀkÃ3 ð1 À r a1 Þðk1 þ r1 kT Þ þ nkÃ4 þ ðaaD EÃ1s ÞkÃ5
þ ð1 À ra1 Þðk1 þ rkT ÞkÃ7 Þ;

ABC a Ã
Dtf k4
t

ð24Þ

P 2 ÞhkT ÞkÃ5

þ ðnð1 À P2 ÞhkT ÞkÃ6 þ ð1 À nÞP2 hkT kÃ9 þ nP 2 hkT kÃ10 Þ;
ABC a Ã
Dtf k3
t

À

À
Ã

ÁÀ
a

a

¼ Àk5 1 À r 2 k2 þ r2 hkT

Á

ð27Þ

À
ÁÀ a
ÁÁ
þ nkÃ6 þ kÃ9 1 À r a2 k2 þ r2 hkT ;
ð28Þ

ABC a Ã
Dtf k6
t

a
¼ ððÀkÃ6 ð1 À ra2 Þðk2 þ

a

r2 hkT Þ þ l

a
Þ þ kÃ10 ð1 À r a2 Þðk2 þ

b à à b à Ã
b
S k À hS k þ ð1 À nÞð1 À P1 Þ SÃ1 kÃ3
N 1 1 N 2 2
N
b
þ r31 ð1 À da1 Þ RÃ1s kÃ3
N

¼ ð1 À

Àð1 À r a1 Þ

b Ã
b
b

k þ nP2 SÃ2 kÃ10 h þ c11 kÃ11 À r31 ð1 À da1 Þ RÃ1s kÃ11
N 10
N
N

b à Ã
b
b
E k þ nð1 À P1 Þ SÃ1 kÃ4 þ RÃ1R kÃ4 r32 ð1 À da1 Þ
N 1s 3
N
N

b à Ã
R k Þ;
N 2R 14

ð30Þ

be à à be à Ã
be
S k À hS k þ ð1 À nÞð1 À P1 Þ SÃ1 kÃ3
N 1 1 N 2 2
N
À
À
Á
a Á be à Ã
a be à Ã
þ r31 1 À d1

R k À 1 À r1
E k
N 1s 3
N 1s 3
À
Á
be à à be à Ã
þ nð1 À P1 Þ S1 k4 þ R1R k4 r32 1 À da1
N
N
À
Á be
be
À 1 À ra1 r1 EÃ1R kÃ4 þ ð1 À nÞð1 À P2 Þh SÃ2 kÃ5
N
N
À
À
Á
a Á be à Ã
a
þ r41 1 À d2
R k À 1 À r 2 r2 hkT E2s kÃ5
N 2R 5
À
Á
be
be
þ nð1 À P2 Þh SÃ2 kÃ6 þ kÃ6 hr42 À 1 À r a2 r2 hkT EÃ2R kÃ6
N

N
À
Á
be
þ ð1 À nÞP1 SÃ1 kÃ7 þ 1 À r a1 r1 hkT E1s kÃ7
N

¼ ð1 À

À
Á
be
a
þ s1 aaD þ c12 þ la þ d þ x2 u2 ðt Þ kÃ8 þ nP1 SÃ1 kÃ8
N
À
Á
be à Ã
a be à Ã
þ 1 À r1
E k r1 þ ð1 À nÞP2 S2 k9 h þ s1 aaD kÃ9
N 1R 8
N
À
Á
à be Ã
a
þ 1 À r 2 r2 hE2s k9
N
À

Á
be
be
þ 1 À r a2 r2 hE2R kÃ10 þ nP2 SÃ2 kÃ10 h þ kÃ10 s1 aaD þ c11 kÃ11
N
N
À
À
Á be à Ã
a Á be à Ã
À r31 1 À d1
R k þ r32 1 À da1
R k
N 1s 11
N 1R 12
À
Á be à Ã
þ r41 1 À da2
R hk þ
N 2s 13

r42 ð1 À da2 Þ

r2 hkT ÞÞ;
ð29Þ

ABC a Ã
Dtf k7
t

b à Ã
b
S k h þ s1 aaD kÃ9 þ ð1 À r a2 Þr2 hE2s kÃ9
N 2 9
N

À
Áb à Ã
À
Áb à Ã
R k þ r41 1 À da2
R hk
þr32 1 À da1
N 1R 12
N 2s 13

À Á
kj T f ¼ 0; ki ; j ¼ 1; 2; 3; :::; 14;

ABC a Ã
Dtf k2
t

b à Ã
S k þ ð1 À r a1 Þ
N 1 8

b à Ã
E k r1
N 1R 8

þg1 nkÃ7 þ ð1 À nÞP2

Moreover,

ABC a Ã
Dtf k1
t

b à Ã
S k þ ð1 À r a1 Þr1 hkT E1s kÃ7
N 1 7

þðs1 aaD þ gn þ c11 þ la þ d1 þ x1 u1 ðtÞÞkÃ7 þ nP1

@Ha
¼
;
@k12
¼

b à à b Ã
S k þ k hr42
N 2 6 N 6

ABC a Ã
Dtf k9
t

be à Ã

R k Þ;
N 2R 14

be1 Ã Ã be1 Ã Ã
be1 Ã Ã
S k À
hS2 k2 þ ð1 À nÞð1 À P1 Þ
S k
N 1 1
N
N 1 3
b
e
b
e
1 Ã Ã
1 Ã Ã
þ r31 ð1 À da1 Þ
R k À ð1 À r a1 Þ
E k
N 1s 3
N 1s 3
be1 Ã Ã be1 Ã Ã
þ nð1 À P1 Þ
S k þ
R k r32 ð1 À da1 Þ
N 1 4
N 1R 4
be1 Ã Ã
be1 Ã Ã

À ð1 À r a1 Þr1
E k þ ð1 À nÞð1 À P2 Þh
S k
N 1R 4
N 2 5
be1 Ã Ã
þ r41 ð1 À da2 Þ
R k À ð1 À r a2 Þr2 hkT E2s kÃ5
N 2R 5

¼ ð1 À

ð31Þ


130

N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

þnð1 À P2 Þh

À
Á
be1 Ã Ã be1 Ã
S k þ
k hr42 À 1 À r a2 r2 hkT E2R kÃ6 þ
N 2 6
N 6

(ii) Transversality conditions:

kÃj ðT f Þ ¼ 0; j ¼ 1; 2; :::; 14:

Á
be1 Ã Ã À
be1 Ã Ã
ð1 À nÞP1
S k þ 1 À r a1 r1 hkT E1s kÃ7 þ nP1
S k þ
N 1 7
N 1 8
À

(iii) Optimality conditions:

Á be1 Ã Ã
be1 Ã Ã
E k r1 þ ð1 À nÞP2
S k h þ s1 aaD kÃ9
1 À r1
N 1R 8
N 2 9
À
Á
À
Á
be1 Ã
a
þ g2 n þ c21 þ la þ d2 þ x3 u3 ðtÞ kÃ9 þ 1 À r a2 r2 hEÃ2s
k

N 9
À
Á
be1 Ã
be1 Ã Ã
þ 1 À r a2 r2 hEÃ2R
k þ nP2
S k h þ kÃ10 s1 aaD
N 10
N 2 10
À
Á be1 Ã Ã
À
Á be1 Ã Ã
À r31 1 À da1
R k þ r32 1 À da1
R k
N 1s 11
N 1R 12
a

be1 Ã Ã
be1 Ã Ã
þr41 ð1 À d2 Þ
R hk þ r42 ð1 À da2 Þ
R k Þ;
N 2s 13
N 2R 14
a

ABC a Ã
Dtf k10
t

À

¼ ð1 À

þr31 1 À d1

Ha ðSÃ1 ;SÃ2 ; EÃ1s ; EÃ1R ; EÃ2s ;EÃ2R ; IÃ1s ; IÃ1R ; IÃ2s ; IÃ2R ; RÃ2s ; RÃ2R ;uÃ1 ; uÃ2 ;uÃ3 ; uÃ4 ; kj Þ
¼

N

À

RÃ1s kÃ3

a

À 1 À r1

Á be2
N

EÃ1s kÃ3

be2 Ã Ã
þ nð1 À P1 Þ

S k
N 1 4

¼ ðr31 ð1 À

d1 ÞkT kÃ3
a

d1 ÞkT kÃ11
a

þ r31 ð1 À
þx
ABC a Ã
Dtf k12
t

þ r32 ð1 À

à Ã
1 u1 k11

À

d1 ÞkT kÃ4

kÃ11 ðda11

a

¼ ðr41 ð1 À
a

kÃ14 nÞ;

þnþl Þþ
ABC a Ã
Dtf k14
t

þ

þ

À ðd22 þ l

a

ÞkÃ14

a

a

ð34Þ

ð35Þ
à Ã
3 u3 k13

À

kÃ13 ðda12
ð36Þ

þ

uÃ4 kÃ14 Þ

uÃ4 ¼ minf1; maxf0;

ðx4 IÃ2R ÞðkÃ14 À kÃ10 Þ
gg:
B4

ð43Þ

Proof. We find the co-state system Eqs. (24)–(37), from Eq. (21),
where

HÃa ¼ IÃ1s þ IÃ1R þ IÃ2s þ IÃ2R þ
þ

B1 2
B2
B3
u ðtÞ þ u22 ðtÞ þ u23 ðtÞ
2 1
2
2

ABC
ABC
ABC
B4 2
u ðtÞ þ kÃ1a Dat SÃ1 þ kÃ2a Dat SÃ2 þ kÃ3a Dat EÃ1s
2 4

þ kÃ4a Dat EÃ1R þ kÃ5a Dat EÃ2s þ kÃ6a Dat EÃ2R þ kÃ7ABC Dat IÃ1s
ABC

ABC

ABC

c

þ kÃ8a Dat IÃ1R þ kÃ9a Dat IÃ2s þ kÃ10a Dat IÃ2R s þ kÃ11a Ra1st
ABC

ABC

Ã

ABC

Ã

ABC

ABC

ABC

ð44Þ

ABc a Ã
a Dt S1

¼ Aa À ðla þ aaD þ kT ÞSÃ1 ;

ð45Þ

ABC a Ã
a Dt S2

¼ aaD S1 À ðla þ hkT ÞSÃ2 ;

ð46Þ

À
Á
¼ ð1 À nÞð1 À P 1 ÞkT SÃ1 þ r31 1 À da1 RÃ1s
À
Á
a
À 1 À r a1 ðk1 þ r1 kT ÞEÃ1s À ðn þ aaD þ la ÞEÃ1s ;

ð37Þ

ABC a Ã
a Dt E2s

ABC a Ã
a Dt E2R

À
Á
¼ ð1 À nÞð1 À P 2 ÞhkT S2 þ r41 1 À da2 hkT RÃ2s þ aaD EÃ1s
À
Á a
À 1 À r a2 ðk2 þ r2 hkT ÞEÃ2s n þ la ÞEÃ2s ;

a

ð49Þ

ð50Þ

¼ ð1 À nÞP1 kT SÃ1 þ ð1 À r a1 Þðk1 þ r1 kT ÞEÃ1s þ da11 RÃ1s
a

À ðs1 aD þ g1 n þ c11 þ l þ d1 þ x1 uÃ1 ÞIÃ1s ;
a

ABC a Ã
a Dt I 1R

ð48Þ

À
Á
¼ nð1 À P2 ÞhkT SÃ2 þ nEÃ2s þ r42 1 À da2 hkT RÃ2R
þ aaD EÃ1R À ð1 À r a2 Þðk2 þ r2 hkT ÞEÃ2R À la EÃ2R ;

ABC a Ã
a Dt I 1s

ð47Þ

¼ nð1 À P1 ÞkT SÃ1 þ nEÃ1s þ r32 ð1 À da1 kT RÃ1R Þ À ð1
a

¼ ðr42 ð1 À da2 ÞhkT kÃ6 þ d22 kÃ10 þ r42 ð1 À da2 ÞhkT kÃ14
a

ð42Þ

À r a1 Þðk1 þ r1 kT ÞEÃ1R À ðaaD þ la ÞEÃ1R ;

þ n þ aD þ l Þ

þx

ðx3 IÃ2s ÞðkÃ13 À kÃ9 Þ
gg;
B3

ABC a Ã
a Dt E1R

d11 kÃ7

¼ ðr32 ð1 À da1 ÞkT kÃ4 þ da12 kÃ8 À kÃ11 ðda12 þ x2 uÃ2 kÃ12

kÃ9 d21

uÃ3 ¼ minf1; maxf0;

a

À nk12 þ aD k13 Þ;

d2 ÞhkT kÃ5

ð41Þ

ð33Þ

a

a

ðx2 IÃ1R ÞðkÃ12 À kÃ8 Þ
gg:
B2

ABC a Ã
a Dt E1s

þ aaD þ la Þ þ aaD kÃ14 Þ;
ABC a Ã
Dtf k13
t

uÃ2 ¼ minf1; maxf0;

B1

is the Hamiltonian. Moreover, the condition in Eq. (23)also holds,
and the optimal control characterization in Eqs. (40)–(43) can be
derived from Eq. (22). #
Substituting uÃi , i = 1,2,. . .,4 in (3)-(16), we can obtain the following state system:

Á be2 Ã Ã
À
Á be2 Ã Ã
1 À da1
R k þ r32 1 À da1
R k
N 1s 11
N 1R 12

ABC a Ã
Dtf k11
t

ð40Þ

ABC

À
Á
À
Á
be2 Ã
a
þ 1 À r a2 r2 hEÃ2R
k þ c22 þ la þ d2 þ x4 uÃ4 ðtÞ kÃ10 þ nP2
N 10
be2 Ã Ã
Â
S k h
N 2 10

be2 Ã Ã
be2 Ã Ã
R hk þ r42 ð1 À da2 Þ
R k Þ;
N 2s 13
N 2R 14

À

kÃ7 Þ

þ kÃ12a Dat RÃ1R þ kÃ13a Dat RÃ2s þ kÃ14a Dat RÃ2R ;

À
Á

be2 Ã
be2 Ã
þð1 À nÞP 2
S þ s1 aaD kÃ9 þ 1 À r a2 r2 hEÃ2s
k
N 2
N 9

þr41 ð1 À da2 Þ

ðx

Ã
Ã
1 I 1s Þðk11

gg;

ð32Þ

À
Á
Á be1 Ã Ã
be1 Ã Ã À
þ 1 À r a1 r1 hkT EÃ1s kÃ7 þ nP1
S k þ 1 À ra1
E k r1
N 1 8
N 1R 8

Àr31

HððSÃ1 ; SÃ2 ; EÃ1s ;EÃ1R ; EÃ2s ; EÃ2R ;IÃ1s ; IÃ1R ; IÃ2s ; IÃ2R ; RÃ2s ;RÃ2R ; uÃ1 ; uÃ2 ; uÃ3 ;uÃ4 ; kj Þ;

uÃ1 ¼ minf1; maxf0;

À
Á À
Á be2 Ã Ã
be2 Ã Ã
þ
R k r32 1 À da1 À 1 À r a1 r1
E k
N 1R 4
N 1R 4
À
Á be2 Ã Ã
be2 Ã Ã
S k þ r41 1 À da2
R k
þ ð1 À nÞð1 À P2 Þh
N 2 5
N 2R 5
À
Á
be1 Ã Ã be2 Ã
À 1 À r a2 r2 hkT EÃ2s kÃ5 þ nð1 À P2 Þh
S k þ
k hr42
N 2 6

N 6
À
Á
be2 Ã Ã
À 1 À r a2 r2 hkT EÃ2R kÃ6 þ ð1 À nÞP1
S k
N 1 7

À

min

0 uÃ1 ;uÃ2 ;uÃ3 ;uÃ4 1

ð39Þ

be2 Ã Ã be2 Ã Ã
be2 Ã Ã
S k À
hS2 k2 þ ð1 À nÞð1 À P1 Þ
S k
N 1 1
N
N 1 3

a Á be2

ð38Þ

a

a

a

ð51Þ

¼ nP 1 kT SÃ1 þ ð1 À r a1 Þðk1 þ r1 kT ÞEÃ1R þ g1 nIÃ1s þ da12 RÃ1R
a

À ðs1 aaD þ ca12 þ la þ d1 þ x2 u2 ÞIÃ1R
a

ð52Þ


131

N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137
ABC a Ã
a Dt I2s

À

ÁÀ
a

¼ ð1 À nÞP2 hkT SÃ2 þ 1 À r 2

Á

a
k21 þ r2 hkT EÃ2s þ s1 aaD IÃ1s
a

a

Ã
3 u3 ÞI 2s ;

þ d21 R2s À ðg2 n þ ca21 þ la þ d2 þ x
ABC a Ã
a Dt I2R

ABC a Ã
a Dt R1s

ABC a Ã
a Dt R1R

s1 aaD I1R þ da22 R2R À ðca22 þ la þ da2 þ x4 uÃ4 ÞIÃ2R ;

ABC a Ã
a Dt R2s

ð56Þ

À
Á
¼ ca21 IÃ2s þ x3 uÃ3 IÃ2s þ aaD RÃ1s À r41 h 1 À da2 kT RÃ2s
À ðd21 þ n þ

ð54Þ
ABC a Ã
a Dt R2R

¼ ca11 IÃ1s þ x1 uÃ1 IÃ1s À r31 ð1 À da1 ÞkT RÃ1s
À ðda11 þ n þ aaD þ la ÞRÃ1s

À ðda12 þ aaD þ la ÞRÃ1R ;

ð53Þ

À
ÁÀ a
Á
¼ nP2 hkT SÃ2 þ 1 À r a2 k2 þ r2 hkT EÃ2R þ g2 nIÃ2s
þ

À
Á
¼ ca21 IÃ1R þ x2 uÃ2 IÃ1R þ nR1s À r32 1 À da1 kT RÃ1R

la ÞRÃ2s ;

ð57Þ

¼ ca22 IÃ2R þ x4 u4 IÃ2R þ nRÃ2s þ aD RÃ1R À r42 hð1
À da2 ÞkT RÃ2R À ðda22 þ la ÞRÃ2R :

ð55Þ

ð58Þ

Numerical techniques for the fractional optimal control model
Let us consider the following general initial value problem:
ABC a
a D yðt Þ

¼ g ðt; yðtÞÞ; yð0Þ ¼ y0 :

ð59Þ

Applying the fundamental theorem of FC to Eq. (59), we obtain

yðtÞ À yð0Þ ¼

1Àa
a
gðt; yðtÞÞ þ
BðaÞ
CðaÞBðaÞ

Z

t

gðh; yðhÞÞðt À hÞaÀ1 dh;

0

ð60Þ
where BðaÞ ¼ 1 À a þ CðaaÞ is a normalization function, and at t nþ1 , we
have

ynþ1 À y0 ¼

CðaÞð1 À aÞ
a
gðt ; yðt n ÞÞ þ
CðaÞð1 À aÞ þ a n
CðaÞ þ að1 À CðaÞ
Â

n Z
X

m¼0

t mþ1

g Á ðtnþ1 À hÞaÀ1 dh;

ð61Þ

tm

Now, gðh; yðhÞÞ will be approximated in an interval [tk, tk+1]
using a two-step Lagrange interpolation method. The two-step
Lagrange polynomial interpolation is given as follows [22]:

P¼

Fig. 2. Numerical simulations of ðS1 þ S2 þ I1s þ I1R þ I2s þ I2R þ E1s þ E1R þ E2s þ
E2R þ R1s þ R1R þ R2s þ R2R Þ=N and a ¼ 1 with control cases using NS2LIM.

gðt m ; ym Þ
gðtmÀ1 ; ymÀ1 Þ
ðh À t mÀ1 Þ À
ðh À tm Þ:
h
h

ð62Þ

Eq. (62), is replaced in Eq. (61), and by performing the same
steps in [22], we obtain

Fig. 3. Numerical simulations of I1s , I1R , I2s and I2R under different values of a with control cases using NS2LIM.


132

N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

ynþ1 À y0 ¼

CðaÞð1 À aÞ
1
gðt ; yðt n Þ þ
CðaÞð1 À aÞ þ a n

ða þ 1Þð1 À aÞCðaÞ þ a
Â

n
X

a

h gðtm ; yðt m ÞÞðn þ 1 À mÞa

m¼0

ðn À m þ 2 þ aÞ À ðn À mÞa ðn À m þ 2 þ 2aÞ
À h g ðtmÀ1 Þ; yðt mÀ1 ÞÞðn þ 1 À mÞaþ1
a

ðn À m þ 2 þ aÞ À ðn À mÞa ðn À m þ 1 þ aÞ;

ð63Þ

To obtain high stability, we present a simple modification in Eq.
(63). This modification is to replace the step size h with /ðhÞ such
that

 
2
/ðhÞ ¼ h þ O h : 0 < /ðhÞ

1:

For more details, see [54]. Then, the new scheme is called the
nonstandard two-step Lagrange interpolation method (NS2LIM)
and is given as follows:

ynþ1 À y0 ¼

CðaÞð1 À aÞ
1
gðt ; yðt n ÞÞ þ
ða þ 1Þð1 À aÞCðaÞ þ a
CðaÞð1 À aÞ þ a n
Â

n
X

a

/ðhÞ gðt m ; yðt m ÞÞ

m¼0

ðn þ 1 À mÞa ðn À m þ 2 þ aÞ À ðn À mÞa ðn À m þ 2 þ 2aÞ
a

À /ðhÞ gðtmÀ1 ; yðtmÀ1 ÞÞ
ðn þ 1 À mÞaþ1 ðn À m þ 2 þ aÞ À ðn À mÞa ðn À m þ 1 þ aÞ:

ð64Þ

Then, we use the new scheme in Eq. (64) to numerically solve
the state system in Eqs. (45)–(58), and we use the implicit finite
difference method to solve the co-state system Eqs. (24)–(37) with
the transversality conditions in Eq. (38).
Numerical simulations
In this section, we present two new schemes in Eqs. (63) and
(64) to numerically simulate the fractional- order optimal system
in Eqs. (45)–(58) and Eqs. (24)–(37) with the transversality condition in Eq. (38) using the parameters given in Table 1 and
/ðhÞ ¼ Q ð1 À eÀh Þ, where Q is a positive number less than or equal
to 0.01. The initial conditions are S1 ð0Þ ¼ 8741400, S2 ð0Þ ¼ 200000,
E1s ð0Þ ¼ 557800, E1R ð0Þ ¼ 7800, E2s ð0Þ ¼ 4500, E2R ð0Þ ¼ 3000,
I1s ð0Þ ¼ 20000;
I1R ð0Þ ¼ 2000,
I2s ð0Þ ¼ 1800,
I2R ð0Þ ¼ 800,
R1s ð0Þ ¼ 8000, R1R ð0Þ ¼ 800, R2s ð0Þ ¼ 200, andR2R ð0Þ ¼ 100. For
computational purposes, we use MATLAB on a computer with the
64-bit Windows 7 operating system and 4 GB of RAM. We now
show some numerical aspects of the simulation of the proposed
model in Eqs. (3)–(16). Fig. 2 shows that the summation of all
the unknown of variables in the proposed model in Eqs. (3)–(16)
is strictly constant during the studied time in the controlled case
when using the scheme in Eq. (64). This result indicates that the
proposed method is efficient. Fig. 3 shows the numerical solutions
of I1s , I1R , I2s and I2R using the scheme in Eq. (64) when T f ¼ 200 in
the controlled case. We note that the solutions for different values
of a vary close to the integer-order solution, i.e., the FO model is a
generalization of the integer-order model and the FOCP systems
and is more suitable for describing the real world. In Figs. 4–6,
we examined the numerical results of I1s , I1R , I2s and I2R in the case

a ¼ 0:95; 1, and we note that there are fewer infected individuals

Fig. 4. Numerical simulations of I1s , I1R , I2s and I2R with a ¼ 0:95 and b ¼ 9 without
control cases using NS2LIM.


N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

133

Fig. 5. Numerical simulations of I1s , I1R , I2s and I2R when B1 ¼ B2 ¼ B3 ¼ B4 ¼ 100 and a ¼ 0:95, b ¼ 9 with control cases using NS2LIM.

Fig. 6. Numerical simulations of I1s , I1R , I2s and I2R when B1 ¼ B3 ¼ 5000, B2 ¼ B4 ¼ 100, b ¼ 8, and a ¼ 1 with and without control cases using NS2LIM.

in the control case. These results agree with the results given in
Table 2. Fig. 7 illustrates the behaviour of relevant variables from
the proposed model in Eqs. (3)–(16) for different avalues using
the scheme in Eq. (64). We note that the relevant variables change

under different values of a following the same behaviour. Fig. 8
shows the behaviours of the relevant variables from the proposed
model in Eqs. (3)–(16) for a ¼ 0:8 using the scheme in Eq. (63). We
note that the relevant variables exhibit the same behaviour. Fig. 9


134

N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

Table 2

Comparison of the values of the objective function system using NS2LIM and T f ¼ 50
with and without control cases.
JðuÃ1 ; uÃ2 ; uÃ3 ; uÃ4 Þ with control

a
1

8:7371 Â 10

JðuÃ1 ; uÃ2 ; uÃ3 ; uÃ4 Þ without controls

5

1:0721 Â 106

0.98

5

8:6240 Â 10

1:0581 Â 106

0.95

8:4617 Â 105

1:0383 Â 106

0.90

8:2138 Â 10

5

6

0.80

7:8340 Â 105

9:6373 Â 105

0.75

5

7:7330 Â 10

9:5414 Â 105

0.60

8:2733 Â 105

1:0502 Â 106

Table 3 shows a comparison of the two proposed schemes in Eqs.
(64) and (63) under different values of a with the control case.
The solutions for the scheme in Eq. (64) appear to be slightly more

accurate than those for the scheme in Eq. (63).

Conclusions

1:0082 Â 10

shows the behaviour of the control variables u2 and u3 at different
values of a and T f ¼ 200 using NS2LM. We note that the control
variables exhibit the same behaviour in the integer and fractional
cases. Fig. 10 shows that the proposed scheme in Eq. (64) is more
stable than the scheme in Eq. (63). Table 2 shows a comparison of
the value of the objective function system using Eq. (64) with and
without control cases when T f ¼ 50 and under different values of
a. We note that the values of the objective function system with
the control cases are lower than the values of the objective function system without the controls for all values of 0:6 < a 1.

In this article, an optimal control for a fractional TB infection
model that includes the impact of diabetes and resistant strains
is presented. The fractional derivative is defined in the ABC sense.
The proposed mathematical model utilizes a non-local and nonsingular kernel. Four optimal control variables, u1 , u2 , u3 and u4 ,
are introduced to reduce the number of individuals infected. It is
concluded that the proposed fraction-order model can potentially
describe more complex dynamics than can the integer model and
can easily include the memory effects present in many realworld phenomena. Two numerical schemes are used: 2LIM and
NS2LIM. Some figures are given to demonstrate how the
fractional-order model is a generalization of the integer-order
model. Moreover, we numerically compare the two methods. It is
found that NS2LIM is more accurate, more efficient, more direct
and more stable than 2LIM.

Fig. 7. Numerical simulations of the relevant variables with control cases when B1 ¼ B3 ¼ 500, B2 ¼ B4 ¼ 100 and b ¼ 5 with different values of a using NS2LIM.


N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

Fig. 8. Dynamics of relevant variables of the system in Eqs. (45)–(58) when B1 ¼ B2 ¼ B3 ¼ B4 ¼ 100 and b ¼ 5; with control cases using 2LIM.

Fig. 9. Numerical simulations of the control variables using NS2LIM.

135


136

N.H. Sweilam et al. / Journal of Advanced Research 17 (2019) 125–137

Fig. 10. Numerical simulations of R1 when B1 ¼ B2 ¼ B3 ¼ B4 ¼ 100 and a ¼ 0:9, h ¼ 1 with control case using NS2LIM and 2LIM:

Table 3
Comparison of 2LIM
h ¼ 0:1 and b ¼ 5:

and

NS2LIM

in

the

controlled

case

with

T f ¼ 10,

Variables

2LIM

NS2LIM

a

I1R
I2s

6.0500 Â 103
1.7822 Â 103

1.9694 Â 103
1.5554 Â 103

0.8

I1R
I2s

4.0922 Â 103
3.1513 Â 103

1.9382 Â 103
1.6662 Â 103

0.7

I1R
I2s

2.9203 Â 103
6.2551 Â 103

1.9168 Â 103
2.3815 Â 103

0.6

Compliance with Ethics Requirements
This article does not contain any studies with human or animal
subjects.
Conflict of interest
The authors have declared no conflict of interest.
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